Isaac Newton: The Last Sorcerer. Michael White

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СКАЧАТЬ a tree and see an apple fall and it was this, combined with a wealth of other factors, that inspired his theory. But it is also quite likely that the apple story was a later fabrication, or at least an exaggeration designed for a specific purpose – almost certainly to suppress the fact that much of the inspiration for the theory of gravity came from his subsequent alchemical work.

      On a prosaic level, Newton’s work as an alchemist was completely anathema to the traditional world of science and to society in general. But, beyond that, attempting to transmute base metals into gold – a preoccupation of the alchemists – was a capital offence. Even in old age Newton was determined to maintain his duplicity, both to protect himself and to preserve unsullied his hard-won image as the greatest scientist who had ever lived.

      So, if these stories are false, how then did Newton arrive at the inverse square law for gravity – the first major development towards the elucidation of universal gravitation, the principle that all masses attract all other masses?

      The first step was to use his mathematical studies in order to mould a set of general mathematical principles that he could use to investigate planetary motion. Since his earliest inquiries into basic mathematics, begun two years earlier, during the spring of 1664, he had managed to assimilate the entire canon of known mathematics and then to extend it into totally uncharted waters – ‘For in those days I was in the prime of my age of invention,’ Newton said of himself sixty years later.¹³ He was familiar with the latest work on the mathematics of curves and the principle that tangents can approximate to the curve and allow certain calculations to be managed, but, like many mathematicians of the time, he wanted something more precise. In particular, he was interested in finding the area under a curve (the area between a curve and the x-axis) and a precise value for the curvature (or gradient) of a curve.

      Scholars are in general agreement that the greatest influence upon Newton’s own thinking about these problems came from his reading Descartes’s Geometry during the summer of 1664, but others have pointed out that Isaac Barrow had also made some considerable progress with the mathematics of gradients and curves and that Newton may have learned a great deal from both men.

      During 1665 and early 1666 Newton worked on these problems and devised a method of finding the exact gradient of a curve by a method which has since become known as differentiation. To understand this method we must first recall that a graph is a way of representing a set of values describing a situation. In the last chapter, the example of a ball falling from the tower was used to illustrate how a real situation can be described graphically. Equally, an algebraic equation is another way of describing a situation. In fact a graphical representation and an algebraic description can both represent the same thing. This means that the algebraic and graphical representations are paired, so manipulating equations by a suitable method can lead the mathematician to information about the curves these equations mirror.

      Newton’s greatest mathematical breakthrough was the realisation that a particular manipulation of a suitable equation could lead to a precise value for the gradient of the curve represented by that equation. This method of manipulation is the essence of differentiation. Another process carried out on a equation (a process since named integration) leads the mathematician to the area under a curve represented by that equation. The calculus is the overall term for these two processes of differentiation and integration, and together they are powerful tools for the mathematician and the scientist.

      Although sometimes placed in his ‘Woolsthorpe period’, work on this development was actually begun while Newton was still in Cambridge. By his own account, he had begun to develop the calculus as early as February 1665.¹⁴ His first mathematics paper, dealing with a mathematical process called the summation of infinitesimal arcs of curves (a major step towards a full realisation of the techniques involved in the calculus), was composed in May 1665.

      Once he had a general method for the calculus, the next step was to apply it to the practical problem of planetary motion – how planets orbit the Sun, and the Moon the Earth, and how mathematical laws can represent these movements.

      A thought experiment familiar to natural philosophers was that of the stone on a string. This may be visualised by imagining a stone attached to a string being whirled around the experimenter’s head. In this model, one force draws the stone towards the centre of the circular path and another pulls the stone away. The Dutchman Christiaan Huygens called the first of these forces the centripetal force and the other the centrifugal force. The stone continues to travel in a circle around the experimenter’s head because the two forces cancel out. If the string is cut the stone will fly off in a straight line at a tangent to the circle.

      Using this as a basis, Newton created a thought experiment to determine a way of calculating the outward or centrifugal force experienced by an object travelling in circular motion. To begin with, he imagined a ball travelling along the four sides of a square inscribed in a circle.

      He was able to calculate the force with which the ball struck one of the points of the circle (say, A), and by multiplying this by four (for the sum of the sides of the square) he arrived at a value for the force exerted by the ball in one circuit around the square. But a square is a poor approximation to a circle, and to arrive at closer values for the force the object would exert if it was travelling in circular motion Newton imagined polygons with increasing numbers of sides inscribed in the circle. The more sides the polygon possessed, the closer it came to describing the circle.

      Using principles derived from his own recent mathematical developments, he eventually obtained a value for the force exerted by an object completing a single truly circular revolution.

      From this calculation he could determine the force with which an object travelling in circular motion pulls away from the centre of the circle and hence the relationship between the force and the size of the circle (or orbit). Applying this to rotation of planets around the Sun, he concluded that ‘the endeavour to recede from the Sun will

       Figures 4 and 5.

      be reciprocally as the squares of their distances from the Sun’.^15*

      This means that the distance and the force experienced by a planet receding from the Sun (or the Moon from the Earth) are related by an inverse square relationship. In other words, if planet A orbits the Sun at a certain distance, another planet B (of equal mass) orbiting at twice the distance will experience a receding force one quarter the value of planet A. If another planet of equal mass, planet C, orbits at a distance three times greater than A it will experience a receding force only one ninth the size of the force experienced by planet A.

      In keeping with his lifelong habit of working with whatever materials were at hand, Newton began this thought process on the back of an old lease his mother had used some time earlier.¹⁶ Surviving to this day, the piece of parchment shows a muddled collection of jottings and calculations which, although describing work that eventually led to a law of gravitation, at this stage (1666) merely illustrate his contention that planets experience a receding force from the Sun governed by an inverse square relationship. It was only later that Newton was able to equate this receding force with a force pulling planets towards the Sun and to realise that this pulling force is also governed by the same inverse square law.

      Yet the thought that there existed an equal and opposite force which countered the receding force could not have been far from his thoughts – not least because of the familiar example of the stone СКАЧАТЬ